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\begin{equation*}S=2 \pi \left( c^2 + \frac{bc^2}{\sqrt{a^2-c^2}} F(\theta, m) + b\sqrt{a^2-c^2} E(\theta, m) \right)\frac{a^2(b^2-c^2)}{b^2(a^2-c^2)}\end{equation*}
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